Answer
See answer below
Work Step by Step
Assume that:
$v_1=1+\alpha x^2$
$v_2=1+x+x^2$
$v_3=2+x$
Then we have:
$c_1v_1+c_2v_2+c_3v_3=0$
$c_1(1+\alpha x^2)+c_2(1+x+x^2)+c_3(2+x)=0$
We obtain:
$\begin{bmatrix}
0 & 1 & 2\\
0 & 1 & 1 \\
\alpha & 1 & 0
\end{bmatrix} \approx\begin{bmatrix}
0 & 1 & 2\\
0 & 1 & 1 \\
0 & \alpha -1 & 2\alpha
\end{bmatrix} \approx \begin{bmatrix}
1 & 1 & 2\\
0 & 1 & 1 \\
0 & 0 & -\alpha -1
\end{bmatrix}$
To span $P_2$:
$\rightarrow -\alpha -1 \ne 0$
$\rightarrow \alpha \ne -1$
The set of vectors $\{v_1,v_2,v_3\}$ span $P_2$ if $\alpha \ne-1$
